Operations on Matrices
• Let A = [aij] m x n and B = [bij] m x n be two matrices of same order m x n. Then the sum of the two matrices A and B is defined as a matrix
C = [cij] m x n where cij = aij + bij for all possible values of i and j.
• If A and B are matrices of same order, then A + B = B + A. This is commutative property of matrices under addition. Addition of matrices also holds the associative law.
• For any matrix A, there exists a null matrix O of the same order such that A + O = A = O + A.
• For any matrix A, there exists a matrix – A such that:
• A + (– A) = O = (–A) + A
• Two matrices are subtracted by subtracting corresponding elements of these matrices.
• Let A = [aij] m x n is a matrix and k is a scalar, then kA is another matrix, which is obtained by multiplying each element of A by a scalar k.
• The product AB of two matrices A and B is defined if numbers of columns of A is equal to the number of rows of B.
• Matrix multiplication is not commutative in general.
• Matrix multiplication is associative whenever both sides of equality are defined.
• Matrix multiplication is distributive over matrix addition.
• For every square matrix A, there is an identity matrix of same order such that IA = AI = A.
• The product of two matrices can be null matrix while neither of them is a null matrix. If A is an m × n matrix and O is a null matrix, then
• the product of matrix A with the null matrix O is always a null matrix.
• In case of matrix multiplication if AB = O, then it does not necessarily imply that BA = O.
• The transpose of a matrix is obtained by interchanging its rows and columns. It is denoted by A' or AT.
• Properties of Transpose:
• For any square matrix A with real number entries, A + AT is a symmetric matrix and A – AT is a skew symmetric matrix.
• Any square matrix can be expressed as the sum of a symmetric and a skew symmetric matrix.
Keywords: Addition of matrices, Multiplication of matrices, Difference of matrices, Transpose of a matrix, Multiplication of a matrix by a scalar, Symmetric matrix, Skew symmetric matrix
If A and B are two matrices such that A + B and AB are both defined, thenMarks:1
A and B are square matrices of same order.
Explanation:In square matrices, both addition and multiplication are defined. Therefore, A and B are square matrices of same order.
If A is a square matrix, then AAT is always aMarks:1
(AAT )T = (AT )T AT = AAT
Hence AAT is a symmetric matrix.
If A and B are symmetric matrices of the same order, then (AB – BA) isMarks:2
A skew-symmetric matrix
Let A and B be symmetric.
Then At = A and Bt = B
(AB – BA)t = (AB)t – (BA)t
= BtAt – AtBt = BA – AB = – (AB – AB)
Hence, (AB – BA) is skew-symmetric.
If A is skew-symmetric, then A3 is :Marks:1
Let A be skew symmetric
Then, At = – A
(A3)t = (AAA)t = At. At. At
= (–A) (–A) (–A) = – A3
Hence, A3 is skew-symmetric